Question

The coefficient of x⁴ in ${\left(\frac{x}{2}-\frac{3}{x^2}\right)}^{10}$ is
(a) $\frac{405}{256}$

(b) $\frac{504}{259}$

(c) $\frac{450}{263}$

(d) none of these

Answer 1

(a) $\frac{405}{256}$

$$\begin{gathered} \mathrm{Suppose}{x}^4\mathrm{occurs}\mathrm{at}\mathrm{the}(r+1)\mathrm{th}\mathrm{term}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}. \\ \mathrm{Then},\mathrm{we}\mathrm{have} \\ {T}_{r+1}={}^{10}C_r(\frac{x}{2}{)}^{10-r}{\left(\frac{-3}{2x^2}\right)}^r \\ =(-1{)}^r{}^{10}C_r\frac{3^r}{2^{10-r}}{x}^{10-r-2r} \\ \mathrm{For}\mathrm{this}\mathrm{term}\mathrm{to}\mathrm{contain}{x}^4,\mathrm{we}\mathrm{must}\mathrm{have}: \\ 10-3r=4 \\ \Rightarrow r=2 \\ \therefore \mathrm{Required}\mathrm{coefficient}={}^{10}C_2\frac{3^2}{2^8}=\frac{10\times 9\times 9}{2\times 2^8}=\frac{405}{256} \end{gathered}$$